Bernoulli's Equation

Bernoulli's EquationAs the fluid moves through a pipe of varying cross section and height , the pressure will change along the pipe. Bernoulli's equation is a fundamental equation in fluid dynamics that relates pressure to fluid speed & height. In deriving Bernoulli's equation , we assume that the fluid is incompressible, non viscous and flows in a steady state manner.

DerivationLet us consider the flow of fluid through the pipe in time t as shown in the fig. The force on the upper end of the fluid is P1A1 . The work done on the fluid in moving it through the distance Dx1 is W1 = F1Dx1 = P1A1 Dx1 1P=F/A F=PA

Similarly the work done on the lower end is W2 = -F 2Dx2 = -P2A2Dx2 ..2 The work done in this case is ve because the force on fluid is opposite to that of F1

From the equation of continuity A1v1= A2 v2 Hence A1v1t = A2v2t = V From equation 4 we get

W=P1A1v1t-P2A2v2tW= (P1 P2 )V 5 If m is mass &r is density then V= m /r then W= (P1-P2) m /r .............6 Part of this work is utilized by the fluid in changing its K.E & a part is used in changing its gravitational P.E.

An interesting exampleA stream of air passing over a tube dipped in a liquid will force the liquid to rise in the tube as shown. This effect is used in perfume bottles & paint sprayers.P + r v2 +r g h=constant

Chimney worksA chimney works best when it is tall and exposed to air currents , which reduces the pressure at the top and force the upward flow of smoke.

Applications of Bernoulli's Equation

TERRCELLI,S THEOREMv1

Suppose a large tank of fluid has two small orifices A & B on it. Let us find the speed with which the water flows from the orifice A .

v1

As the orifices are so small efflux speeds v2 and v3 will be much larger than the speed v1 of the top surface of the water. Due to this reason, we can take v1 approximately equal to zero. Now Bernoulli's equation can be written as . P1+rgh1 = P2+1/2 rv22 +rgh2

The top level of the tank has moved down a little and the P.E has been transferred into K.E of the efflux of fluid. If the level of orifice B has the level same as the water of the tank , the water will rise to the level of the water tank.

Relation b/w speed & pressure of the fluid

A result of Bernoulli's equation is that the pressure will be low where the speed of the fluid is high. Suppose that the water flows through a pipe of system as shown in the fig.

It is clear that the water will flow faster at point B than A or C. Assume that the speed at point A is 0.20ms-1 & at B 20ms-1. In order to compare the pressure at B with A, we use Bernoulli, s equation . PA + rvA2 = PB + rvB2 ( P.E is constant at both points)

Putting v A = 0.20 ms-1 v B = 2.0ms-1 & r= 1000kgm-3

PA + rvA2 = PB + rvB2 PA PB = 1980 Nm-2 It means that the pressure in narrow pipe where streamlines are closer together is much smaller than in wider pipe.

RESULTWhere the speed is high , the pressure will be low.

Example1

The wing of aero plane

is designed to deflect the air so that the streamlines are closer together above the wing than below it. Where the streamlines are closer to each other, the speed is faster. The air is traveling faster on the upper side of the wing than the lower.

Example2 When a tennis ball is hit by a racket in such a

way that it spins as well as moves forward , the velocity of the air on one side of the ball increases due it spins and air pressure decreases. This gives extra curvature to the ball & as a result ball swing which deceives an opponent player.